| Step | Hyp | Ref
| Expression |
| 1 | | vex 3484 |
. . . 4
⊢ 𝑤 ∈ V |
| 2 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑣 = 𝑎 → (𝑣 · 𝑏) = (𝑎 · 𝑏)) |
| 3 | 2 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑣 = 𝑎 → (𝑧 = (𝑣 · 𝑏) ↔ 𝑧 = (𝑎 · 𝑏))) |
| 4 | 3 | rexbidv 3179 |
. . . . . 6
⊢ (𝑣 = 𝑎 → (∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 · 𝑏))) |
| 5 | 4 | cbvrexvw 3238 |
. . . . 5
⊢
(∃𝑣 ∈
𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 · 𝑏)) |
| 6 | | eqeq1 2741 |
. . . . . 6
⊢ (𝑧 = 𝑤 → (𝑧 = (𝑎 · 𝑏) ↔ 𝑤 = (𝑎 · 𝑏))) |
| 7 | 6 | 2rexbidv 3222 |
. . . . 5
⊢ (𝑧 = 𝑤 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 · 𝑏) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑤 = (𝑎 · 𝑏))) |
| 8 | 5, 7 | bitrid 283 |
. . . 4
⊢ (𝑧 = 𝑤 → (∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑤 = (𝑎 · 𝑏))) |
| 9 | | supmul.1 |
. . . 4
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏)} |
| 10 | 1, 8, 9 | elab2 3682 |
. . 3
⊢ (𝑤 ∈ 𝐶 ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑤 = (𝑎 · 𝑏)) |
| 11 | | supmul.2 |
. . . . . . . . . . 11
⊢ (𝜑 ↔ ((∀𝑥 ∈ 𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥))) |
| 12 | 11 | simp2bi 1147 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
| 13 | 12 | simp1d 1143 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| 14 | 13 | sselda 3983 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ) |
| 15 | 14 | adantrr 717 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ ℝ) |
| 16 | | suprcl 12228 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈
ℝ) |
| 17 | 12, 16 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈
ℝ) |
| 18 | 17 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → sup(𝐴, ℝ, < ) ∈
ℝ) |
| 19 | 11 | simp3bi 1148 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)) |
| 20 | 19 | simp1d 1143 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ⊆ ℝ) |
| 21 | 20 | sselda 3983 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ ℝ) |
| 22 | 21 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ ℝ) |
| 23 | | suprcl 12228 |
. . . . . . . . 9
⊢ ((𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥) → sup(𝐵, ℝ, < ) ∈
ℝ) |
| 24 | 19, 23 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → sup(𝐵, ℝ, < ) ∈
ℝ) |
| 25 | 24 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → sup(𝐵, ℝ, < ) ∈
ℝ) |
| 26 | | simp1l 1198 |
. . . . . . . . . . 11
⊢
(((∀𝑥 ∈
𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)) → ∀𝑥 ∈ 𝐴 0 ≤ 𝑥) |
| 27 | 11, 26 | sylbi 217 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 0 ≤ 𝑥) |
| 28 | | breq2 5147 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑎 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑎)) |
| 29 | 28 | rspccv 3619 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝐴 0 ≤ 𝑥 → (𝑎 ∈ 𝐴 → 0 ≤ 𝑎)) |
| 30 | 27, 29 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑎 ∈ 𝐴 → 0 ≤ 𝑎)) |
| 31 | 30 | imp 406 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 0 ≤ 𝑎) |
| 32 | 31 | adantrr 717 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 0 ≤ 𝑎) |
| 33 | | simp1r 1199 |
. . . . . . . . . . 11
⊢
(((∀𝑥 ∈
𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)) → ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) |
| 34 | 11, 33 | sylbi 217 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) |
| 35 | | breq2 5147 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑏 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑏)) |
| 36 | 35 | rspccv 3619 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝐵 0 ≤ 𝑥 → (𝑏 ∈ 𝐵 → 0 ≤ 𝑏)) |
| 37 | 34, 36 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑏 ∈ 𝐵 → 0 ≤ 𝑏)) |
| 38 | 37 | imp 406 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 0 ≤ 𝑏) |
| 39 | 38 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 0 ≤ 𝑏) |
| 40 | | suprub 12229 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
| 41 | 12, 40 | sylan 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
| 42 | 41 | adantrr 717 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
| 43 | | suprub 12229 |
. . . . . . . . 9
⊢ (((𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥) ∧ 𝑏 ∈ 𝐵) → 𝑏 ≤ sup(𝐵, ℝ, < )) |
| 44 | 19, 43 | sylan 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ≤ sup(𝐵, ℝ, < )) |
| 45 | 44 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ≤ sup(𝐵, ℝ, < )) |
| 46 | 15, 18, 22, 25, 32, 39, 42, 45 | lemul12ad 12210 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
))) |
| 47 | 46 | ex 412 |
. . . . 5
⊢ (𝜑 → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
| 48 | | breq1 5146 |
. . . . . 6
⊢ (𝑤 = (𝑎 · 𝑏) → (𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )) ↔ (𝑎 · 𝑏) ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
| 49 | 48 | biimprcd 250 |
. . . . 5
⊢ ((𝑎 · 𝑏) ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )) → (𝑤 = (𝑎 · 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
| 50 | 47, 49 | syl6 35 |
. . . 4
⊢ (𝜑 → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑤 = (𝑎 · 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
))))) |
| 51 | 50 | rexlimdvv 3212 |
. . 3
⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑤 = (𝑎 · 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
| 52 | 10, 51 | biimtrid 242 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐶 → 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
| 53 | 52 | ralrimiv 3145 |
1
⊢ (𝜑 → ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
))) |